Related papers: Statistics on $\ell$-interval parking functions
Finding an available on-street parking spot is a relevant problem of day-to-day life. In recent years, cities such as Melbourne and San Francisco deployed sensors that provide real-time information about the occupation of parking spots.…
We explore the link between combinatorics and probability generated by the question "What does a random parking function look like?" This gives rise to novel probabilistic interpretations of some elegant, known generating functions. It…
The problem of reservation in a large distributed system is analyzed via a new mathematical model. A typical application is a station-based car-sharing system which can be described as a closed stochastic network where the nodes are the…
Place a car independently with probability $p$ at each site of a graph. Each initially vacant site is a parking spot that can fit one car. Cars simultaneously perform independent random walks. When a car encounters an available parking spot…
Initially a car is placed with probability p at each site of the two-dimensional integer lattice. Each car is equally likely to be East-facing or North-facing, and different sites receive independent assignments. At odd time steps, each…
We address optimal placement of vehicles with simple motion to intercept a mobile target that arrives stochastically on a line segment. The optimality of vehicle placement is measured through a cost function associated with intercepting the…
Classical parking functions are a generalization of permutations that appear in many combinatorial structures. Prime parking functions are indecomposable components such that any classical parking function can be uniquely described as a…
We analyze properties of the firing map, which iterations give information about consecutive spikes, for periodically driven linear integrate-and-fire models. By considering locally integrable (thus in general not continuous) input…
In a scenario of growing usage of park-and-ride facilities, understanding and predicting car park occupancy is becoming increasingly important. This study presents a model that effectively captures the occupancy patterns of park-and-ride…
We introduce a new parking procedure called MVP parking in which $n$ cars sequentially enter a one-way street with a preferred parking spot from the $n$ parking spots on the street. If their preferred spot is empty, they park there.…
A classical parking function of length $n$ is a list of positive integers $(a_1, a_2, \ldots, a_n)$ whose nondecreasing rearrangement $b_1 \leq b_2 \leq \cdots \leq b_n$ satisfies $b_i \leq i$. The convex hull of all parking functions of…
We study subadditive functions of the random parking model previously analyzed by the second author. In particular, we consider local functions $S$ of subsets of $\mathbb{R}^d$ and of point sets that are (almost) subadditive in their first…
We propose a generalized car parking problem where either a car of size $\sigma$ or of size $m\sigma$ ($m>1$) is sequentially parked on a line with probability $q$ and $(1-q)$, respectively. The free parameter $q$ interpolates between the…
The choice of forward and reverse parking in a parking lot is studied as a stochastic process. An $M/M/c/c$ queueing system is used as an initial framework. We use Monte Carlo simulation to get the relationship between vehicle orientation…
Consider the following simple parking process on $\Lambda_n := \{-n, \ldots, n\}^d,d\ge1$: at each step, a site $i$ is chosen at random in $\Lambda_n$ and if $i$ and all its nearest neighbor sites are empty, $i$ is occupied. Once occupied,…
The \emph{Shi arrangement} is the set of all hyperplanes in $\mathbb R^n$ of the form $x_j - x_k = 0$ or $1$ for $1 \le j < k \le n$. Shi observed in 1986 that the number of regions (i.e., connected components of the complement) of this…
We consider the notion of classical parking functions by introducing randomness and a new parking protocol, as inspired by the work presented in the paper ``Parking Functions: Choose your own adventure,'' (arXiv:2001.04817) by Carlson,…
We show that the notion of parkization of a word, a variant of the classical standardization, allows to introduce an internal product on the Hopf algebra of parking functions. Its Catalan subalgebra is stable under this operation and…
We initiate the study of the cycle structure of uniformly random parking functions. Using the combinatorics of parking completions, we compute the asymptotic expected value of the number of cycles of any fixed length. We obtain an upper…
An asymmetric exclusion process type process, where cars move forward along a closed road that starts and terminates at a parking garage, displays dynamic phase transitions into two types of condensate phases where the garage becomes…